{
  "abstract": "Denote by $ω(n)$ the number of distinct prime divisors of the natural number $n$. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments $\\sum_{n\\leq x}(ω(n)-\\log\\log x)^{k}$, for a wide range of integers $k\\geq 2$. In this notes, we shall apply the method for $ω_{z}(n)$ which denotes the number of distinct prime divisors $\\leq z$ of $n$. Especially, for odd integers $k\\geq 3$, we lead asymptotic formulas for $\\sum_{n\\leq x}(ω_{z}(n)-\\log\\log z)^{k}$, under certain restrictions on $k$, $z$, and $x$. Also, we refer to a framework of the approach.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22777",
  "author_total": 3,
  "authors": [
    {
      "name": "T. Makoto Minamide"
    },
    {
      "name": "Haruka Sakai"
    },
    {
      "name": "Yoshio Tanigawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/95yqqgdy0eh2f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22777",
  "fulltext_url": null,
  "id": "95yqqgdy0eh2f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22777",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214030343",
  "pdf_url": "https://arxiv.org/pdf/2609.22777",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/95yqqgdy0eh2f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22777",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22777"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On a Turán's theorem for small primes",
  "title_text": "On a Turán's theorem for small primes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/95yqqgdy0eh2f",
  "venue": "arXiv",
  "versions": []
}